The 35-year-old has signed a two-year deal at the Riverside after agreeing personal terms and passing a medical.
Gareth Southgate: "Danny is an excellent goalkeeper who will push for a first team place this season.
"He has played at the highest level with Wales and has great experience. Just as importantly, he is the right type of character in that he is very professional, trains well and has a real hunger to claim the number one spot.
"He played 40 games for Tranmere last season, kept 20 clean sheets and was probably the best goalkeeper in League One.
"We feel we need an experienced goalkeeper to challenge Brad [Jones] and Jason [Steele]. Danny has a wealth of experience and would no doubt have played many more times but for injury."